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GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS
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GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

Jan 14, 2016

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Page 1: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

GEOMETRY CONSTRUCTIONSCREATING PERFECT SHAPES WITHOUT NUMBERS

Page 2: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

TO ERR IS HUMAN

In mathematics and science when we prove ideas, we are trying to show that it works for every case. To do this we can’t just pick numbers and try it once, but it’s impossible to try it for every number.

Also, we can’t use human measurements of length and degrees. This is because humans aren’t perfect and the tools we use aren’t perfect either.

Page 3: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

MATH WITHOUT THE NUMBERS

There are some proofs of geometry concepts that can be done with formal constructions.

A formal geometric construction uses only a compass and a straight edge. This removes the use of human measurement and generalizes the proof to work for all possible measures.

Page 4: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

TOOLS OF THE TRADE

A Geometry compass is tool that is used for drawing arcs and circles. The most important part of choosing a compass is making sure it will hold it’s position.

Anything with a rigid straight side can be used for a straight edge. I like to use popsicle sticks instead of rulers because they don’t have any numbers to tempt students to cheat on their constructions

Page 5: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CONSTRUCTION 1

A perpendicular bisector of a line segment, divides the segment into two equal parts with a perpendicular line.

This construction also can be used to find the midpoint of a line segment.

PERPENDICULAR BISECTOR

Page 6: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

PERPENDICULAR BISECTOR (MIDPOINT)

Page 7: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CONSTRUCTIONS 2 AND 3

We’ll use the same idea and techniques to make a perpendicular that isn’t the bisector. For the first one the point is on the line segment. For the second one the point is not on the line segment.

PERPENDICULAR THROUGH A POINT

Page 8: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

PERPENDICULAR LINE THROUGH POINT ON LINE SEGMENT

Page 9: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

PERPENDICULAR LINE THROUGH POINT NOT ON SEGMENT

Page 10: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CONSTRUCTION 4

An angle bisector divides an angle into two equal angles.

ANGLE BISECTOR

Page 11: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

ANGLE BISECTOR

Page 12: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CONSTRUCTION 5

To make congruent angles we need to make a copy of an angle.

COPYING AN ANGLE

Page 13: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

MAKING CONGRUENT ANGLES PART 1

Page 14: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

MAKING CONGRUENT ANGLES PART 2

Page 15: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CONSTRUCTION 6

An equilateral triangle has three congruent sides and three congruent angles.

EQUILATERAL TRIANGLE

Page 16: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

EQUILATERAL TRIANGLES

Page 17: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CONSTRUCTION 7

Circum means around and scribe means write. So to circumscribe a circle around a triangle means to draw a circle around the triangle.

To do this we will be making

perpendicular bisectors of

the sides of the triangle.

CIRCUMSCRIBING

Page 18: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CIRCUMSCRIBING A CIRCLE AROUND A TRIANGLE

Page 19: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CONSTRUCTION 8

This time we’re going to construct a circle inside the triangle.

INSCRIBING

Page 20: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

INSCRIBING A CIRCLE IN A TRIANGLE

Page 21: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

CONSTRUCTION 9

Because circles have 360 degrees they are easily separated into six segments. This is why hexagons are often found in nature, like basalt columns, honeycombs and crystals.

HEXAGONS

Page 22: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

HEXAGONS

Page 23: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

HEXAGONS

Page 24: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

HEXAGONS

Page 25: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

HEXAGONS

This method can also be used for making an equilateral triangle By just connecting every other mark.

Page 26: GEOMETRY CONSTRUCTIONS CREATING PERFECT SHAPES WITHOUT NUMBERS.

WHAT ELSE CAN YOU MAKE WITH YOUR COMPASS?

Flower of Life drawing from one of Leonardo DaVinci’s sketchbooks.